Author | Riesel, Hans. author |
---|---|
Title | Prime Numbers and Computer Methods for Factorization [electronic resource] / by Hans Riesel |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1985 |
Connect to | http://dx.doi.org/10.1007/978-1-4757-1089-2 |
Descript | XVI, 464 p. online resource |
1. The Number of Primes Below a Given Limit -- 2. The Primes Viewed at Large -- 3. Subtleties in the Distribution of Primes -- 4. The Recognition of Primes -- 5. Factorization -- 6. Prime Numbers and Cryptography -- Appendix 1. Basic Concepts in Higher Algebra -- Modules -- Euclidโs Algorithm -- The Labour Involved in Euclidโs Algorithm -- A Definition Taken from the Theory of Algorithms -- A Computer Program for Euclidโs Algorithm -- Reducing the Labour -- Binary Form of Euclidโs Algorithm -- Groups -- Lagrangeโs Theorem. Cosets -- Abstract Groups. Isomorphic Groups -- The Direct Product of Two Given Groups -- Cyclic Groups -- Rings -- Zero Divisors -- Fields -- Mappings. Isomorphisms and Homomorphisms -- Group Characters -- The Conjugate or Inverse Character -- Homomorphisms and Group Characters -- Appendix 2. Basic Concepts in Higher Arithmetic -- Divisors. Common Divisors -- The Fundamental Theorem of Arithmetic -- Congruences -- Linear Congruences -- Linear Congruences and Euclidโs Algorithm -- Systems of Linear Congruences -- Carmichaelโs Function -- Carmichaelโs Theorem -- Appendix 3. Quadratic Residues -- Legendreโs Symbol. -- Arithmetic Rules for Residues and Non-Residues -- The Law of Quadratic Reciprocity -- Jacobiโs Symbol -- Appendix 4. The Arithmetic of Quadratic Fields -- Appendix 5. Continued Fractions -- What Is a Continued Fraction? -- Regular Continued Fractions. Expansions -- Evaluating a Continued Fraction -- Continued Fractions as Approximations -- Euclidโs Algorithm and Continued Fractions -- Linear Diophantine Equations and Continued Fractions -- A Computer Program -- Continued Fraction Expansions of Square Roots -- Proof of Periodicity -- The Maximal Period-Lenath -- Short Periods -- Continued Fractions and Quadratic Residues -- Appendix 6. Algebraic Factors -- Factorization of Polynomials -- The Cyclotomic Polynomials -- Aurifeuillian Factorizations -- Factorization Formulas -- The Algebraic Structure of Aurifeuillian Numbers -- Appendix 7. Multiple-Precision Arithmetic -- Various Objectives for a Multiple-Precision Package -- How to Store Multi-Precise Integers -- Addition and Subtraction of Multi-Precise Integers -- Reduction in Length of Multi-Precise Integers -- Multiplication of Multi-Precise Integers -- Division of Multi-Precise Integers -- Input and Output of Multi-Precise Integers -- A Complete Package for Multiple-Precision Arithmetic -- Appendix 8. Fast Multiplication of Large Integers -- The Ordinary Multiplication Algorithm. -- Double Length Multiplication. -- Recursive Use of Double Length Multiplication Formula -- A Recursive Procedure for Squaring Large Integers -- Fractal Structure of Recursive Squaring -- Large Mersenne Primes -- Appendix 9. The Stieltjes Integral -- Functions With Jump Discontinuities -- The Riemann Integral -- Definition of the Stieltjes Integral -- Rules of Integration for Stieltjes Integrals -- Integration by Parts of Stieltjes Integrals -- The Mean Value Theorem -- Applications -- Tables -- Tatle 1. The Primes Belnw 12553 -- Table 4. Factors of Fermat Numbers -- Table 7. Factors of Mersenne Numbers -- Table 32. Quadratic Residues -- Table 33. Gaussโ formulas for Cyclotomic Polynomials -- Table 34. Lucasโ formulas for Cyclotomic Polynomials